722 PROCEEDINGS OF SECTION J. 



definite area. He learns to measure, to calculate, to generalise, to 

 operate upon algebraical symbols.* At the same time he learns the 

 most important geometrical facts. I The nature of loci is taught as 

 greatly facilitating the generalisation of geometrical truths. Further, 

 he learns the theory of limits and the nature of symmetry, which are 

 both powerful means of understanding geometrical truths. As a good 

 instance of generalisation we may consider the deduction of many 

 useful facts from the theorem established experimentally that the 

 locus of points equidistant from two given points is the perpendicular 

 bisector of their join. By the aid of this we may co-ordinate the usual 

 methods of drawing perpendiculars, bisecting angles and straight 

 lines and combined with the idea of limits the drawing tangents to 

 circles, &c. The circular measure of angles should be considered here 

 as furnishing the basis for the practical construction of equal angles, 

 drawing parallels, &c. The nature of limits is well brought home by 

 inscribing a square in a circle, bisecting tlie quadrants, and re-bisecting 

 until further bisection is beyond the power of practical construction. 

 The pupil learns to regard a tangent as the limit of a chord. All this 

 is quite within the range of the average boy when treated experimentally 

 and not merely formally ; and he prepares himself well for the later 

 work and the application of mathematics. 



Then the solution of equations is dealt with, and the equations to 

 be solved are plotted on squared paper. I am accustomed to have 

 all the mathematical work done in squared paper books, with the 

 exception of mere arithmetical computation, which is made in a 

 scribbling book, and the results recorded in the squared paper book. 

 The pupil learns how to plot a straight line, and to regard the linear 

 equation ?/ — ax + 6 as the symbolical representation of simple pro- 

 portion. He learns experimentally the nature of the trigonometrical 

 tangent, the relation between the trigonometrical tangents of per- 

 pendiculars and parallels, and draws perpendiculars and parallels in 

 this way ; how to calculate the length of the join of two points, the 

 co-ordinates of the mid point of the join, and other facts of the analytical 

 geometry of the straight line. Suitable physical data are supplied 



* The inductive method is ever kept prominent. We want a student, for 

 example, to understand the geometrical nature of a prism. A large number of 

 .solids are shown the pupil, cryst allographic models being serviceable on account 

 ■of their variety. The teacher picks out and groups together all prisms. The 

 pupil examines them, to find out their distinguishing characters. If his answer is 

 too comprehensive he may be shown other solids which, on his saying, would be 

 included in this group ; on the other hand, if his distinction is too restricted, some 

 ■of the solids placed in the group would be excluded. When the jiupil has learnt 

 in tliis way how to distinguish prisms from other solids, the prisms may be divided 

 by the teacher into two groups — right and oblicpie. The pupil is to discover the 

 basis of this differentiation. Then, again, the right prisms may be divided into 

 square, rectangular, hexagonal, &c. The knowledge gained by the class in this 

 way is far deeper than that gained by a didactic treatment of the definition of a 

 prism, though this would occupy only a few minutes. 



t No time is wasted over any proof that the opposite vertical angles of two 

 intersecting straight lines are equal. The pons asinorum vanishes into thin air 

 as the pupil realises the symmetry of an isosceles triangle. 



